Chernoff approximations as a way of finding the resolvent operator with applications to finding the solution of linear ODE with variable coefficients

The method of Chernoff approximation is a powerful and flexible tool of functional analysis that in many cases allows expressing exp(tL) in terms of variable coefficients of linear differential operator L. In this paper we prove a theorem that allows us to apply this method to find the resolvent of operator L. We demonstrate this on the second order differential operator. As a corollary, we obtain a new representation of the solution of an inhomogeneous second order linear ordinary differential equation in terms of functions that are the coefficients of this equation playing the role of parameters for the problem.